Summary: We develop an adaptive \({\mathcal H}_\infty\) control approach for a class of polynomial nonlinear systems with parametric uncertainties. Motivated by the dissipation theory and the vector projection technique, we propose a nonlinear adaptive \({\mathcal H}_\infty\) controller and its associated parameter adaptation law. The proposed adaptive control strategy is capable of identifying unknown parameter values quickly and minimizing the effect of estimation error. To further improve adaptive controlled performance, the Lyapunov function will be relaxed from quadratic to higher-order forms and the controller gains are generalized from constant to parameter-dependent. All of the synthesis conditions are formulated in the framework of polynomial/constant linear matrix inequalities and solvable using available software packages.